Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Find the differential dy of the given function.
![**Problem 15:**
Find the function for \( y \) given by the equation:
\[ y = \frac{x + 1}{2x - 1} \]
This rational function can be analyzed for various characteristics, including its domain, asymptotes, and behavior near critical points.
- **Domain:** The function is undefined when the denominator equals zero. Solve \( 2x - 1 = 0 \) to find the restriction: \( x = \frac{1}{2} \). So, the domain is \( x \in \mathbb{R} \setminus \{\frac{1}{2}\} \).
- **Vertical Asymptote:** The line \( x = \frac{1}{2} \) is a vertical asymptote, as the function approaches infinity or negative infinity near this value.
- **Horizontal Asymptote:** As \( x \) approaches infinity, the horizontal asymptote can be determined by the ratio of the leading coefficients in the polynomial numerator and denominator. Thus, \( y = \frac{1}{2} \) is the horizontal asymptote.
Educators can use this function to teach about rational expressions, graphing, asymptotes, and analyzing function behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa66fd4df-0aa7-4b7b-bbe2-d7f92ff0b206%2F5139cf9c-986e-4b68-8e74-5db4827b320d%2Fm0kqoqo_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 15:**
Find the function for \( y \) given by the equation:
\[ y = \frac{x + 1}{2x - 1} \]
This rational function can be analyzed for various characteristics, including its domain, asymptotes, and behavior near critical points.
- **Domain:** The function is undefined when the denominator equals zero. Solve \( 2x - 1 = 0 \) to find the restriction: \( x = \frac{1}{2} \). So, the domain is \( x \in \mathbb{R} \setminus \{\frac{1}{2}\} \).
- **Vertical Asymptote:** The line \( x = \frac{1}{2} \) is a vertical asymptote, as the function approaches infinity or negative infinity near this value.
- **Horizontal Asymptote:** As \( x \) approaches infinity, the horizontal asymptote can be determined by the ratio of the leading coefficients in the polynomial numerator and denominator. Thus, \( y = \frac{1}{2} \) is the horizontal asymptote.
Educators can use this function to teach about rational expressions, graphing, asymptotes, and analyzing function behavior.
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